Volume 2013, Article ID 478389,18pages http://dx.doi.org/10.1155/2013/478389

### Research Article

### The Effect of Uncertainty in the Excitation on

### the Vibration Input Power to a Structure

### A. Putra

1### and B. R. Mace

21_{Faculty of Mechanical Engineering, Universiti Teknikal Malaysia Melaka, Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, Malaysia}

2_{Department of Mechanical Engineering, University of Auckland, Auckland 1142, New Zealand}

Correspondence should be addressed to A. Putra; azma.putra@utem.edu.my

Received 30 April 2013; Accepted 18 July 2013

Academic Editor: Marc homas

Copyright © 2013 A. Putra and B. R. Mace. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In structural dynamic systems, there is inevitable uncertainty in the input power from a source to a receiver. Apart from the nondeterministic properties of the source and receiver, there is also uncertainty in the excitation. his comes from the uncertainty of the forcing location on the receiver and, for multiple contact points, the relative phases, the force amplitude distribution at those points, and also their spatial separation. his paper investigates quantiication of the uncertainty using possibilistic or probabilistic approaches. hese provide the maximum and minimum bounds and the statistics of the input power, respectively. Expressions for the bounds, mean, and variance are presented. First the input power from multiple point forces acting on an ininite plate is examined. he problem is then extended to the input power to a inite plate described in terms of its modes. he uncertainty due to the force amplitude is also discussed. Finally, the contribution of moment excitation to the input power, which is oten ignored in the calculation, is investigated. For all cases, frequency band-averaged results are presented.

### 1. Introduction

he treatment of structure-borne sound sources remains a challenging problem. Structural excitation to a building loor, for example, by active components like pumps, compressors, fans, and motors, is an important mechanism of sound generation. To obtain an accurate prediction of the injected input power from such sources, both the source and the receiver must irstly be characterised. However in practical application, the variability of source and receiver properties including the lack of knowledge in the excitation force creates uncertainty in the input power. he problem is exacerbated because in practice there will usually be multiple contact points (typically four) and 6 degrees of freedom (3 for translation and 3 for rotation) at each, and that force and moment components at each contact point will contribute to the total input power. herefore to assess the uncertainty, some quantiication of the bounds, mean, and variance of the input power is of interest.

he uncertainty in vibrational energy due to random properties, for example, dimensions, shapes, boundary

conditions, and so on in a simple receiver structure, such as a plate, has been described by Langley and Brown [1,2], where expressions for the mean vibrational energy and its variance were developed. A closed form solution was presented for the relative variance as a function of modal overlap factor and the nature of the excitation [1]. In [2], the analysis was extended to the ensemble average of the frequency band-averaged energy as a function of the frequency bandwidth. he analysis proceeded on the assumption that the natural frequencies form a random point process with statistics governed by, for example, the Gaussian orthogonal ensemble (GOE). he same type of analysis for a more complex system has also been proposed [3].

he efective mobility concept was also employed in [8] to estimate the total power to an ininite beam through four con-tact points. he importance of having knowledge of the force distribution at the contact points was acknowledged. For this purpose, three simple force ratio assumptions were intro-duced, and the efects of force position, type of excitation, the structural loss factor, the receiver, and the number of contact points were investigated. It was found that the estimation is accurate only in the mass-controlled region and when the system approximated a symmetrical response. In a later paper [9], the force ratio was deined in terms of its statistical dis-tribution. Simple expressions of the distribution at the mass-, stifness-, and resonance-controlled regions were derived.

Characterisation of a source in practice can be ap-proached by using a reception plate method [10, 11] which is based on laboratory measurements, where from measured mobilities and surface mean-square velocities of the receiver, the free velocities and mobilities of the source can be extracted.

his paper focuses only on the uncertainty in the excita-tion with the source and receiver assumed to be deterministic. he source may have multiple contact points. he uncertainty in input power due to the excitation phase, its location, and separation of the contact points is investigated. First some general comments are made. Broadband excitation is des-cribed, although only time-harmonic excitation is considered here with frequency averages subsequently being taken. he input power from multiple point forces to an ininite plate is examined to give an insight into the physical mechanisms involved. In practice, the receiver will have modes, although the modal overlap might be high. he input power to a inite plate is then analysed, where now the forcing location on the receiver becomes important. he mean and the variance of the input power averaged over force positions are investi-gated. he results are also presented in frequency-band aver-ages.

he uncertainty in the input power due to uncertainty in the force amplitude for multiple contact points is also discussed for a simple case of two contact points. Rather than dealing with force ratios between contact points, the sum of the squared magnitudes of the forces is assumed known.

Finally, the inclusion of moments in the excitation is investigated and predictions of its contribution to the input power are made.

### 2. Input Power and Uncertainty Assessment

2.1. Time-Harmonic Excitation. Consider a vibrating source

connected through a single or�contact points to a receiver. For a time-harmonic excitation at frequency �, the input power is expressed as a function of mobility (or impedance) of the source and receiver [10,12]. his requires knowledge of both source and receiver mobilities and the so-called blocked force or free velocity of the source. In general, the mobilities are matrices and the blocked forces or the free velocities are vectors, with the elements relating to the various translational and rotational degrees of freedoms (DOFs) at the contact points. In this paper, however, the analysis is made by assuming that the force excitation is known, and the

source mobility is assumed to be much smaller than that of the receiver, as is usually the case in practice. he input power is therefore given by

�in= 12Re{̃**F**∗**Y**̃̃**F} ,** (1)

where**F**̃ = [�1���1 �2���2⋅ ⋅ ⋅ ������]�is the vector of the

complex amplitudes of the time-harmonic forces and where

∗denotes the conjugate transpose. he�th force has a real
magnitude�_{�}and phase�_{�}. he mobilities of the receiver are
represented by an�×�matrix**Y**̃. In this section, only forces
are considered. Moment excitation is discussed inSection 6.

2.2. Broadband Excitation. For a broadband excitation over

a frequency band�, the input power should be deined in terms of power spectral density and can be written as

��in = � ∑

� ���

Re{̃�_{��}} +

� ∑ �,� � ̸= �

Re{̃�_{��}}Re{̃�_{��}} , _{(2)}

where �_{��} and ̃�_{��} are the autospectral density and
cross-spectral density of the forces, respectively, and̃�_{��}is the
trans-fer mobility between points�and�. he cross-spectral
densi-ties would oten in practice be diicult to measure. However,
the coherence relates the auto- and cross-spectral densities of
the excitation. hus by assuming that only the autospectra of
the forces are known, the cross-spectra are such that

�����̃�������� = ±√�2������, (3)

where0 ≤ �2 ≤ 1is the coherence. his gives maximum and minimum bounds to the magnitude of the cross-spectral density. A rather similar approach is proposed by Evans and Moorhouse [13] for the case of a rigid body source, where the real part of the cross-spectral density is predicted using the available data of the autospectra and the calculated free velocities at the contact points from the source rigid body modes. Hence, this is limited only to the mass-controlled region of the source at very low frequencies. he comparison with measured data shows a good agreement. However, neither of these approaches are implemented in this paper. All forces are assumed to be time harmonic.

2.3. Uncertainty Quantification. Two approaches are

em-ployed to describe the uncertainty in the input power, namely, possibilistic and probabilistic approaches [14]. he possibilistic approach gives an interval description of the input power, which lies between lower and upper bounds; that is,

�in∈ [�in �in] , (4)

where�inand�inare the minimum and maximum bounds

and�inis the interval variable. One example has been given in

Section 2.2, where the input power can be bounded by using

he probabilistic approach gives information about the likelihood and probability of the input power. he variation is speciied by a probability density functionΠ. IfΠ(�)is a con-tinuous function of some variable�, the mean or the expected value of the input power and its variance are deined by

��in = � [�in] = ∫

��in(�) Π (�)d�,
��in2 = ∫_{�}�in2 (�) Π (�)d� − (��in)

2 .

(5)

### 3. An Infinite Plate Receiver

he input power to an ininite plate, as an ideal simple struc-ture, is irst investigated. he point and transfer mobilities for an ininite plate are given by [15]

̃��= _{8√���}1 ,

̃��= ̃��[�0(2)(��) − 2�_{� �}0(��)] ,

(6)

where�_{0}(2)is the zeroth-order Hankel function of the second
kind,�_{0}is the zeroth-order modiied Bessel function of the
second kind,�is the mass per unit area, and�_{�} = �ℎ3/12(1−

]2) is the bending stifness of the plate having Young’s

modulus �, thickness ℎ, and Poisson’s ratio ]. he input

point mobility is purely real and independent of frequency, behaving as a damper. In the transfer mobility, the irst and second terms represent propagating and near-ield outgoing waves, respectively.

Assume an ininite plate is excited by two point forces
separated by a distance�. From (1), the force vector is**F** =
{�1���1 �2���2}�_{and the mobility is a}2 × 2_{matrix. he total}

power is, thus, the sum of the input power at each location which yields

�in= 12Re{̃��} (�12+ �22) +Re{̃��} �1�2cos�, (7)

where� = �1 − �2is the phase diference between the two
forces. Note that ̃�_{12} = ̃�_{21} = ̃�_{�}. For the case of an ininite
structure, the input mobility is the same at any position which
implies that̃�_{11}= ̃�_{22}= ̃�_{�}.

3.1. Dependence on the Contact Points Separation. When the

structural wavelength�is much larger than the separation
of the excitation points (� ≫ �), the transfer mobility is
approximately equal to the point mobility. herefore,̃�_{�}≈ ̃�_{�}.
For simplicity, if�1= �2= �, the input power becomes

�in=Re{̃��} �2(1 +cos�) . (8)

From (8), the maximum and minimum input power are

�in= 2�2Re{̃��} , for� = 2�� �in= 0, for� = (2� + 1) �,

(9)

where�is any integer.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

kL

N

o

rm

ali

sed po

w

er

10−2 10−1 100 101 102

Figure 1: he normalised input power to an ininite plate subjected

to two in-phase (—) and out-of-phase (⋅ ⋅ ⋅) harmonic unit point

forces (— grey line: max/min bounds at higher frequencies).

When the structural wavelength is much smaller than� (� ≪ �), the input power is

�in= (Re{̃��} +Re{̃��}cos�) �2. (10)

For small structural wavelength, the maximum and mini-mum input powers depend on Re{��}cos�which indicates the dependency of the power on the phase and the distance between the excitation forces. Now since�� ≫ 1, the real part of the transfer mobility in (6) can be expressed in its asymptotic form [16] as (seeAppendix A)

Re{̃�_{�}} =Re{̃�_{�}} √ 2

���cos(�� − �4). (11)

he input power can thus be written as

�in=Re{̃��} (1 + √ 2_{���}cos(�� − �4)cos�) �2. (12)

he input power at higher frequencies is thus bounded by

�in= (1 ± √ 2_{���)}Re{̃��} �2. (13)

Figure 1shows the total input power when the two forces

are in-phase (� = 0) and out-of-phase (� = �) as a function of��. Note that the input power has been normalised with respect to the input power from two point forces acting incoherently (equal to two times the power from a single point force). It can be seen that the power luctuates around the value it would have if the two forces were applied inde-pendently. From (12), the power is minimum and maximum when�� = (2� + 1)�for� = 0, 1, . . .for in-phase and out-of-phase forces, respectively. hese are when, with respect to the wavelength, the two in-phase forces become out of phase and the out of phase forces become in phase. he intersection between the two curves at high��is when� = �/2. For�� <

3.2. Random Phase. In practice, accurate information regard-ing the relative phase between the forces is generally not known. It might be assumed that all possible relative phases have equal probability. he probability density functionΠof the phase is then constant and given by

Π (�) = 1_{2�, � = [0, 2�] .} (14)

From (10), the mean power and its variance using (5) are

��in=Re{̃��} �2,

�2

�in = 12(Re{̃��} �
2_{)}2_{.}

(15)

It can be seen that the mean power equals that which would be given by two forces acting independently. he variance, hence, arises from interference between the two forces. As

̃��= ̃��for�� ≪ 1, from (15), the normalised standard

devia-tion is�/� = 1/√2. For�� ≫ 1,�decreases as��increases (see (11)). Substituting (11) into (15) gives

�2

�2 = ( 1_{���)}cos2(�� − �4). (16)

he bounds of the variance occur when cos2(�� − �/4) = 1. he maximum normalised standard deviation is therefore

�/� = 1/√���.

When the separation distance of the contact points is uncertain, consequently��mod�becomes unknown, while

1/(��)is more or less constant. herefore by averaging (16) over all possible��mod �, the variance can be expressed as

⟨�_{�}2_{2}⟩

��= ( 1���) [ 1 � ∫

�/2

−�/2cos 2

(�� − �4)d(��)]

= 1_{2���.}

(17)

herefore, the normalised standard deviation due to uncer-tainty in��is�/� = 1/√2���.

Figure 2shows the input power for a number of possible

relative phases of excitation. he mean power is a function of the point mobility which then gives a constant value with frequency. he normalised deviation of the power (1 ± �/�) lies between the maximum and minimum input power and bounded by1±1/√���. It can be seen that uncertainty in�� increases the deviation of the input power.

3.3. Four Contact Points. he case of four input excitations is

more realistic in practice, for example, a vibrating machine with four feet.Figure 3shows an ininite plate excited by four harmonic point forces. In this case, from (1), the mobility is a

4 × 4matrix. he diagonal elements are the point mobilities
at each location, which for an ininite structure are equal
to ̃�_{�}. he analysis is generally complicated. For simplicity,
by assuming that the excitation positions form a rectangular

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

kL

N

o

rm

ali

sed po

w

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10−2 10−1 100 101 102

Figure 2: he normalised input power to an ininite plate subjected to two harmonic unit point forces with various relative phases (grey

lines):− −mean,− ⋅ −mean±standard deviation, — (thick lines)

max/min bounds, — mean±bounds of standard deviation, and⋅ ⋅ ⋅

mean±bounds of standard deviation due to uncertainty in��.

h

F3ej�3

F4ej�4

F2ej�2

F1ej�1

L2

L3

L1

Figure 3: An ininite plate excited by four harmonic point forces applied at the vertices of a rectangle.

shape, therefore, ̃�_{12} = ̃�_{21} = ̃�_{34} = ̃�_{43}; ̃�_{13} = ̃�_{31} = ̃�_{24} =

̃�42and ̃�14 = ̃�41 = ̃�23 = ̃�32. he problem can be

simpli-ied numerically by choosing a force vector whose phases are relative to the phase of one reference force. he force vector can be rewritten as

̃

**F**= [�1 �2���2 �3���3 �4���4]�, _{(18)}

where��is the relative phase of�_{�}with respect to�1. hus,
the total input power is given by

�in= 12Re{̃��} (�12+ �22+ �32+ �42)

+Re{̃�_{12}} (�_{1}�_{2}cos�_{2}+ �_{3}�_{4}cos(�_{3}− �_{4}))

+Re{̃�13} (�1�3cos�3+ �2�4cos(�2− �4))

Note again in (19), that the input power is the sum of the powers that would be injected by the forces acting individu-ally (the irst term) and terms that depend on the relative phases of the forces (the remaining terms).

hus, if equal probability of relative phase is assumed and the phases are uncorrelated, the probability density function

Πcan be expressed as

Π (�2, �3, �4) = Π (�2) Π (�3) Π (�4) ,

Π (��) = 1_{2�.}

(20)

Using (5) and assuming all the forces have equal amplitudes, the mean power and the variance are given by

��in = 2Re{̃��} �

2_{,} _{(21)}

�2�in = 12(Re{̃�12} �
2_{)}2

+ 12(Re{̃�13} �2)2

+ 12(Re{̃�14} �2)2

+ 12(Re{̃�23} �2)2+ 12(Re{̃�24} �2)2

+ 12(Re{̃�_{34}} �2)2.

(22)

As in the case of two point forces inSection 3.2, the variance of the input power depends only on the transfer mobilities while the mean power is only a function of the point mobility. In general, the mean and the variance of input power averaged over all possible phases of excitation for�contact points for a rectangular separation can be written as

��in= 12 � ∑

�

Re{̃�_{��}} �_{�}2,

��in2 = 12 �−1

∑ �=1

� ∑ �=2 �<�

(Re{̃���} ����)2.

(23)

he bounds for the normalised input power are obtained by inserting the maximum transfer mobility from (11) into (19), which for equal force amplitudes yields

�in

� = 1 ±√ 2�� (�−1/21 + �−1/22 + �−1/23 ) . (24)

In the same way, substituting (11) into (22), the bounds for the normalised standard deviation are

�

� = ±√ 12��(�−11 + �−12 + �−13 )1/2, (25)

where the positive and negative signs are for the maximum and minimum bounds, respectively.

Following the same method for the case of two point forces, the bounds due to uncertainty in��are given by

�

� = ±0.5√ 1��(�−11 + �−12 + �−13 )1/2. (26)

0 0.5 1 1.5 2 2.5 3 3.5 4

N

o

rm

ali

sed po

w

er

10−1 100 101

kL1

Figure 4: he normalised input power to an ininite plate subjected to four harmonic unit point forces with various phases (grey lines):

− −mean,−⋅−mean±standard deviation, — (thick lines) max/min

bounds, — mean±bounds of standard deviation, and⋅ ⋅ ⋅mean±

bounds of standard deviation due to uncertainty in��.

Figure 4 shows the input power with various relative

phases of excitation. As in Figure 2, the power luctuates
at high��around the mean value. he standard deviation
shown is from (22). he irst dip in the standard deviation
corresponds to the frequency where the diagonal�_{3}equals
half the structural wavelength, and the subsequent dips are
related to �1 and �_{2}. Equations (25) and (26) again show
that the standard deviation reduces as the separation of the
contact points increases.

### 4. Input Power to a Finite Plate

For a inite structure, the mobility and, hence, the input power can be expressed in terms of a summation over modes of vibration. he mobility of a inite structure at an arbi-trary point (�, �) subjected to a point force�at (�0, �0) at frequency�is given by [10]

̃� = ��∑∞ �=1

Φ�(�0, �0) Φ�(�, �) �2

�(1 + ���) − �2 , (27)

where Φ_{�} is the �th mass-normalised mode shape of the
structure,��is the�th natural frequency, and��is damping
loss factor of the�th mode. A case oten considered is that of a
rectangular plate with simply supported boundary conditions
as this system provides a simple analytical solution. For a
simply supported rectangular plate with dimensions�×�, the
mode shape and the natural frequency for mode(�, �)are

Φ��(�, �) = 2

√�sin(���� )sin(���� ) ,

���_{= √ �� [(}��_{� )}2+ (��_{� )}2] ,

where�is the total mass of the plate and�, � = 1, 2, 3, . . .. he rigid body motion existing in the reception plate method [11] can be approximated by assuming the edges are guided or sliding. he mode shape functions involve cos terms instead of sin terms in (28). Using the corresponding real part of the point mobility in (27), the input power for point excitation is given by

4.1. Averaging over Force Positions. Equation (29) implies

that the input power depends on the forcing location which might be uncertain.Figure 5shows the input power from a single point force for various forcing locations. he structure considered is an aluminium plate having dimensions0.6 ×

0.5 × 0.0015m, Young’s modulus 7.1 × 1010N/m2, � =

2700kg/m3, and damping loss factor� = 0.05, assumed con-stant for every mode. he input power is normalised with respect to the input power to an ininite plate.

he peaks occur at the resonances of the plate. hey are distinct at low frequencies, but the modal overlap increases as the frequency increases. Analytically, since

1/(��) ∫_{�}∫_{�}Φ2

��d�0d�0 = 1/�, the mean input power

aver-aged over all possible force positions is [10] as follows:

⟨�in⟩_{�}_{0}_{,�}_{0}= |�|

It can be seen that at high frequencies, the mean power converges to the same level as the input power to an ininite plate. he variance of the input power also decreases at high frequencies as with respect to the spatial variation of the vibration modes, the forcing location becomes less important as the frequency increases.

4.2. Averaging over Frequency Bands. Suppose that the

exci-tation frequency lies between two frequencies�_{1}and�_{2}. he
input power can then be averaged over this frequency band
and can be expressed as

⟨�in⟩_{�}= _{�2}_{− �1}1 ∫
�1

�2 �in(�)

d�. (31)

Figure 6shows the average input power for a 100 Hz

band-width and at 50 Hz centre frequency spacing. For the plate example considered, the modal density is 0.065 modes/Hz. herefore with this bandwidth, there are on average just over 6 modes in the band. It can be seen that the mean value is now close to the ininite plate value except below 100 Hz, where the modal overlap is low and the response is stifness dominated.

4.3. Prediction of Mean and Variance. he frequency average

of the input power in (29) strongly depends on the statistical
distribution of�/((�2_{��}− �2)2+ (�_{��}2 �)2). For a symmetric
structure like a rectangular plate, asymptotically the natural
frequency spacings have a Poisson distribution; that is, the

Frequency (Hz)

Figure 5: he normalised input power to a inite plate subjected to a single harmonic point force for various possible force positions (grey

lines): — mean and− ⋅ −mean±standard deviation.

Frequency (Hz)

Figure 6: he normalised input power to a inite plate subjected to a single harmonic point force for various possible force positions

averaged over 100 Hz frequency band (grey lines): — mean and−⋅−

mean±standard deviation.

spacings between successive natural frequencies are statisti-cally independent and have an exponential distribution [17]. From [1,2,10,17], the mean and variance of the input power averaged over all possible forcing locations yield

⟨�in⟩_{(�,�}_{0}_{,�}_{0}_{)}= |�|

0 0.5 1 1.5 2

No

rm

al

is

ed

p

o

w

er

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0 5 10

MOF

Frequency (Hz)

(a)

0 0.5 1 1.5 2

No

rm

al

is

ed

p

o

w

er

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0 5 10

MOF

Frequency (Hz)

(b)

Figure 7: he mean and standard deviation of the input power of a inite plate due to variation in forcing locations (numerical: — mean,−⋅−

mean±standard deviation, and (32):− −mean and⋅ ⋅ ⋅mean±standard deviation; (a)� = 0.05and (b)� = 0.1).

Figure 7 shows the normalised standard deviation of

the input power averaged over force positions for diferent damping loss factors together with the modal overlap factor (MOF), where MOF = ����. It can be seen that there is a good agreement between the numerical calculations and the analytical predictions if MOF> 1.

From (32), the relative standard deviation can be deined as a function of modal overlap factor; that is,

��= ⟨�⟩_{⟨�}
in⟩

=_{√����� =}1 1

√� MOF. (33) For plates with other shapes, the natural frequency spac-ing statistics are not Poisson (e.g., under many circumstances they asymptote to Gaussian orthogonal ensemble statistics). An alternative expression for the variance can then be found [1,2].

4.4. Four Point Excitations. In this section, results are

pre-sented for the case where there are four rectangularly dis-tributed point excitations. he diagram of the force positions is the same as in Figure 3 for an ininite plate. he total input power is also similar to (19) except that for a inite structure the leading term is now expressed in terms of the input mobility for each force location. herefore by also assuming equal force amplitudes and equal probability of all the excitation phases, the mean power for a inite plate subjected to four point forces according to (23) is given by

��in = 12 (Re{̃�11} +Re{̃�22} +Re{̃�33}

+Re{̃�44}) �2,

(34)

whilst the variance is the same as in (22).

Figure 8 shows the normalised mean and the standard

deviation of the input power to a plate having dimensions

0.2 1 10 30

N

o

rm

ali

sed po

w

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10−2

10−3

10−1

100

101

102

kL1

Figure 8: he normalised input power to a inite plate subjected to four harmonic point forces averaged over all possible excitation

phases (— mean and− ⋅ −mean±standard deviation).

1 × 0.8 × 0.0015m. he simulation is made for forces with a square distribution, where�1 = �2 = 0.1m (see Figure

3), and is located around the middle of the plate. he result shows typical behaviour where the variation is larger at low frequencies and decreases as��increases.

0 5 10 15 20 25 −25

−20 −15 −10 −5 0 5

Frequency (Hz)

r�

(dB r

e 1)

(a)

0 5 10 15 20 25

−25 −20 −15 −10 −5 0 5

Frequency (Hz)

r�

(dB r

e 1)

(b)

Figure 9: he relative standard deviation of input power to a inite plate subjected to four harmonic point forces averaged over various force

positions and frequency bands: — numerical,− ⋅ −ininite plate and− −(33) ((a)� = 0.1and (b)� = 0.05).

the excitation points is a multiple of half the structural wavelength,�� = ��. However,Figure 9(b)shows that for a smaller damping, the ininite plate result is now an under-estimate of the numerical values. In this case, the prediction using (33) gives a good agreement, particularly at large��.

### 5. Random Force Amplitude

Besides the phases and locations of excitation, the force distribution at the contact points also generates variability in the input power. In practice, this could be due to the condi-tion of the installacondi-tion or the nature of the source itself. he problem is that it is very diicult in practice to have know-ledge of its distribution in suicient detail. he situation becomes more complicated still if moment excitation is also taken into consideration. hus, in theoretical studies oten, for convenience, only translational forces of equal amplitude are assumed [8,11].

Here, the approach is used to constrain the total square magnitude of the forces, that is, to assume

�ef2 = � ∑ �=1�

2

� (35)

is constant. he goal is to obtain the maximum and minimum bounds of the mean and variance of the input power provid-ing that only the so-called efective force�efis known. From

(23) for the mean input power to an ininite plate excited by two point forces,

��in = 12 ̃��(� 2

1+ �22) = 12 ̃���ef2. (36)

It can be seen that for an ininite plate, the mean power does not depend on the force distribution but rather on the charac-teristics of the receiver only. However, the variance depends on the product of the forces at the contact points. For two

forces, the maximum variance is obtained when both contact points have equal force distribution; that is,�1 = �2, and the minimum variance is when one of the forces equals the efective force while the other is zero; that is,�1= 0;�2= �ef

(see (23)). For a inite plate, the location of the excitation
determines the mobility of the receiver. However, if the spatial
variation is small (�� < �), the mean input power also
depends only on the receiver mobility as in (36), where ̃�_{�}=

̃�11≈ ̃�22.

Figure 10(a) shows the distribution of mean and the

standard deviation of the input power due to random relative
phases for various force amplitudes in frequency averages,
where�_{ef}2 = 1N2. he excitation points are located at(0.25�,

0.6�)and(0.58�, 0.6�) for the same plate dimensions as in

Section 4.1with damping loss factor� = 0.01. It can be seen

that at low frequency, the variation is up to 4 dB which is quite signiicant. However, this reduces as the frequency increases. he variation can also be reduced by increasing the damping loss factors as shown inFigure 10(c).

he average mean and variance from this results can be
obtained by assuming�_{1}2is uniformly probable between 0 and

�2

ef. Within this range, the probability density functionΠcan

be assumed constant; that is,

Π (�2
1) = 1_{�}2

ef ,

�2

1 = [0, �ef2] .

(37)

Using (5), the mean and the variance of the input power due to random phases averaged over random point force loading for the case of two contact points are given by

⟨��in⟩_{�}2

1 = 14 (Re{̃�11} +Re{̃�22}) �

2 ef,

⟨�2

�in⟩�_{1}2= 112(Re{̃�12} �
2
ef)

2 .

0 5 10 15 20 25 30 35

P

o

w

er (dB r

e 1 wa

tt)

−30

−25 −20 −15

kL

(a)

0 5 10 15 20 25 30 35

P

o

w

er (dB r

e 1 wa

tt)

−30

−25 −20 −15

kL

(b)

0 5 10 15 20 25 30 35

P

o

w

er (dB r

e 1 wa

tt)

−30

−25 −20 −15

kL

(c)

0 5 10 15 20 25 30 35

P

o

w

er (dB r

e 1 wa

tt)

−30

−25 −20 −15

kL

(d)

Figure 10: (a, c) he distribution of the mean (dark grey) and mean±standard deviation (light grey) of the input power to a inite plate with

two contact points due to random relative phases for various force amplitudes; (b, d) he average mean (—) and average mean±standard

deviation (− ⋅ −) over random force amplitudes. (a)-(b)� = 0.01; (c)-(d)� = 0.1.

Figures 10(b) and 10(d) show the average mean and standard deviation of the input power from the results in Figures 10(a) and 10(c). Here, again it can be clearly seen that the deviation decreases with frequency and also as the damping loss factor increases. In this example, for� = 0.1, the deviation of the power from the mean is insigniicant above

�� = 5(<1 dB).

Figure 11 shows the mean and variance due to random

phases and force amplitudes for the ininite plate, also with
two contact points, where in (38),̃�_{11}= ̃�_{22}= ̃�_{�}. Asymptotic
form of̃�_{�}in (11) is used to calculate the maximum and
mini-mum bounds (the case where�_{1}2 = �_{2}2 = �ef2/2). he range

between the bounds can be seen to be roughly 2 dB between

�� = 4 and 16. However, this decreases as the frequency increases.

### 6. The Contribution of Moment Excitation

6.1. he Efect of Moment Excitation on the Input Power. In the

previous sections, only the translational force is considered as the driving excitation. However, in principle the motion at a contact point of a structure-borne source would involve up to six components where not only forces, but also moments will contribute to the total input power. he moment excitation is oten neglected partly because of measurement diiculties rather than the fact that, in most cases, it gives a small contribution to the input power [18]. he contribution of moments is most important at higher frequencies and is less important than that of forces if the source is far away from discontinuities or boundaries [19].

Figure 12 illustrates the components of excitation

0 2 4 6 8 10 12 14 16 18

the input power to an ininite plate with two contact points due to random relative phases and random force amplitudes (— max-min

mean±standard deviation, (thick line) max-min bounds).

point is a function of point mobilities, transfer mobilities for diferent axes, and also the cross-mobilities for diferent components. herefore, there will be a6 × 6mobility matrix for each excitation point. he problem becomes more com-plicated for multiple contact points. For�contact points, the interaction between components will increase the size of the system matrix to6� × 6�.

In this paper, however, the problem is simpliied by neglecting the in-plane excitations, that is,��,��, and��. herefore, the mobility matrix is reduced to a3 × 3matrix for a single contact point. In general, the input power due to a combined point force and moment excitation can be rewritten as

and ̃� ̇��are the point force and point moment mobilities and ̃� ̇�� and ̃�V�

are the cross-mobilities from force to rotation and from moment to translation, respect-ively. Sincẽ� ̇��= ̃�V�

In matrix form, the power can be expressed as in (1) where

̃

Figure 12: Six components of point excitations.

With inclusion of the moment excitation, the mobility matrix for a single contact point is given by

̃

where**Y**̃is symmetric.

6.2. Magnitude of Moment. he relative contribution to the

input power depends of course on the magnitude of the excitation. Force and moment cannot be compared directly as they have diferent units. In a practical situation, they would also depend on the nature of the force generation mechanism in the source. he installation condition also has to be considered. he efects of moment excitation for a vibrating machine installed on sot support at the contact points would be diferent to those if the machine was bolted tightly to the receiver structure. hus, the problem remains of qualifying the relative efects of force and moment. Moorhouse [18] pro-posed a dimensionless mobility where, for example, the real part of a cross-mobility, Re{̃�̇��}, is normalised by the real part of the corresponding point mobilities for both the force and the moment,√Re{̃� ̇��}Re{̃�V�}

. his gives insight into the relative contribution due to the diferent excitation com-ponents.

6.2.1. Single Point Excitation. he relative importance of force

and moment in exciting a structure can be compared only in terms of their input power. However, to calculate the power not only the mobilities should be known but also the magnitudes and the phases of the excitation components (see (39)). Petersson [20] introduced the nondimensional

eccentricitywhich relates the ratio of magnitude of moment

and force to the structural wavenumber.

Here, another approach is introduced where the
magni-tudes of the moments,� = ��̃ ��1_{, and the force,} ̃� = ����2_{,}

at the contact point are related by an efective lever arm�by

� = ��, (42)

relative input power. he total input powers,�_{�}and��, due
to a force and a moment on an ininite plate are

�in= ��+ ��= 12Re{̃�

V�

} �2

+ 12Re{̃�̇��} �2, (43) and the real parts of the point mobilities are given by

Re{̃�V� Consequently, the relative phase between the force and the moment is irrelevant.

From (42), (43), and (44), the input power from moment excitation can be scaled in terms of the input power from the force by a nondimensional unit��and is expressed as

��= (��)2��, (45)

where�is the structural wavenumber. Equation (43) can be rewritten as

�in= ((��)2+ 1) ��. (46)

Figure 13shows the normalised total input power to an

ininite plate for a single contact point. It can be seen that the power from force excitation is constant with frequency while the power from moment excitation is increasing with frequency. Both powers intersect at�� = 1. For�� < 1, the power is dominated by force excitation and for�� > 1, the power is dominated by moment excitation.

For a inite plate receiver, the total input power is given as in (40). While the situation is now numerically complicated, (46) can again be used to scale the individual contribution to the input power.

6.2.2. Multiple Point Excitation. Figure 14shows a diagram

of a translational force�which generates moment �that can be resolved into moments��and��components. he moments can be expressed as

��= ���sin(�) ,

��= −���cos(�) , (47)

where � is the lever arm, or the distance from the line of action of �to the point attached to the structure,�is the angle between the lever arm and the positive�-axis, and� is a dimensionless scaling factor.

Equations (42) and (47) can be used to deine the relation between force and moment for multiple contact points.

Figure 15shows the forces and moments for a typical four

point contact source, with the points having a rectangular
distribution, where�2_{3} = �2_{1} + �2_{2}. he reference moment
at any contact point might then be considered as a sum of
contributions from forces at all the contact points. In this

10−2

Figure 13: he normalised input power from force (− ⋅ −) and

moment (− −) excitations at a single contact point and the total

power (—).

Figure 14: he lever arm of the moment and force.

x

Figure 15: he moment and force directions at source-receiver interface with four contact points.

and the moments about the�-axis are

{ he subsequent sections discuss the efect of moment excita-tion on the input power to ininite and inite plates particu-larly for the multiple point excitation.

6.3. Infinite Plate Receiver

6.3.1. Single Contact Point. For a single contact point,

Figure 13 shows the input power as a function of �� and

the relative phase between the force and moment is not important. However, for multiple contact points, the relative phases are required as the result of the coupling between forces and moments to the response at another contact point.

6.3.2. Multiple Contact Points. As an example, for two contact

points there are iteen relative phases. Assume the distance� between the two points is parallel with�-axis (� = 0) so that some transfer moment mobilities about�-axis become zero. he transfer moment mobilities are given inAppendix A. In this case, the total input power for the case of two contact points is given by

�in= 12Re{̃�

where�_{�}denotes the point mobility (the same contact point)
and�_{�}denotes the transfer mobility (diferent contact point).
he phase � denotes the relative phase between the two
components at the same or diferent contact points; for
example,�4is the relative phase between the moment about
the�-axis and the force at diferent points. In (50), it has
been noted that̃�_{�}V�� = ̃�_{�}̇���. Due to the complexity of this
expression, it is diicult to determine the bounds of input
power analytically. However for simplicity, it is assumed that
all the components are in-phase, so that �� = 0 for � =

1, 2, 3, 4, and5. By also assuming�1 = �2 = �,�1 = �2and following the same method as in (48) and (49) for a2 × 2 matrix, thus,��,1 = ��,2 = �� = ��,��,1 = (� + ��)�, and��,2 = (� − ��)�. he asymptotic forms of the transfer mobility in (50) for this case can be expressed as (see also

Appendix A) the cos and sin terms equal to unity, the maximum and minimum bounds of the input power normalised with respect to the input power from translational force (��) for in-phase excitation are found to be

�in

his reduces to (13), the case where there is only translational force excitation, when�� ≪ 1and��� ≪ 1.

InSection 3.2, assuming random phases with equal

N

Figure 16: he normalised input power to an ininite plate subjected

to two harmonic unit point forces and two harmonic moments:− −

mean, — (thick line) max/min bounds; (52),− ⋅ −mean±standard

deviation and — mean±bounds of standard deviation; (55),⋅ ⋅ ⋅

mean±bounds of standard deviation due to uncertainty in��and

(56) (� = 0.003m,� = 0.003).

to an ininite plate receiver through�contact points can, in general, be expressed as

��in_{= 12}∑�

where � and � indicate the �th and �th contact points, respectively.

he bounds of the normalised standard deviation can be obtained by substituting (51a), (51b), and (51c) into (54). Ater algebraic manipulation, it can be approximated by

� deviation for force excitation (see (16)). Following the same method in(17), the standard deviation due to uncertainty in the dimensionless spacing��is given by

�

Figure 16shows the mean input power and its standard

deviation. he trend is the same as inFigure 2for transla-tional force excitation, except that the input power tends to increase at high frequencies due to moment excitations.

6.4. Finite Plate Receiver

6.4.1. Single Contact Point. For a inite plate, the phase

difer-ence between the force and moment becomes important as the cross-mobility is not zero. For the same plate dimensions as inSection 4.1,Figure 17shows the normalised input power against��for a single contact point assuming in-phase force and moment. he result inFigure 17(a)shows the increase of the input power due to moment contribution at �� >

0.25(see also Figure 13). For the case where the excitation is near to the plate edge inFigure 17(b), the total power is signiicantly less at low��, because the point mobility for force excitation (which dominates at low frequencies,�� ≪

1) is smaller near the edge. However, when�� > 0.2, the input power is the same as that when the excitation position is near to the centre of the plate due to the increasing power from the moment, so that it compensates partly for the reducing power from the force.

Figure 18shows the normalised input power for various

forcing locations on the plate. he increase in the mean power due to the contribution of moment excitation can be seen roughly above�� > 0.35.

Figure 19shows the relative standard deviation�_{�}of the

averaged input power for diferent damping loss factors and magnitudes of moment excitation. For all cases, it can be seen that the relative standard deviation, in an average sense, agrees reasonably well with that from the translational force from (33). his indicates that the ratio between the mean and standard deviation is approximately the same even if the moment excitation is neglected in the calculation of the input power. Particular attention is focused on the results at large

��when the moment starts to contribute substantially to the total input power.

6.4.2. Multiple Contact Points. Again, the relative phases

10−1

Figure 17: he normalised input power of a inite plate subjected to force and moment excitations at a single contact point ((a) the power with

(− −) and without (—) moment and (b) the total power for the contact point around the edge (− −) and middle (—) of the plate:� = 0.005m,

Figure 18: he normalised input power to a inite plate subjected to force and moment excitations at single contact point for various possible

forcing locations (grey lines): — mean and− ⋅ −mean±standard deviation. (a) discrete frequency and (b) frequency-band average;� =

0.005m,� = 0.1.

will produce a displacement at the same point. herefore, the variance is given by

�2 For four contact points, the mobility matrices are 12 ×

0 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k�

−15 −10 −5

r�

(dB r

e 1)

(a)

0 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k�

−15 −10 −5

r�

(dB r

e 1)

(b)

0 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k�

−15 −10 −5

r�

(dB r

e 1)

(c)

Figure 19: he relative standard deviation of the input power to a inite plate subjected to force and moment excitations at single contact point

averaged over various possible forcing locations and frequency bands; — numerical and− −(33) ((a)� = 0.005m,� = 0.1; (b)� = 0.005m,

� = 0.01; and (c)� = 0.01m,� = 0.01).

0 5 10 15 20 25

0 0.5 1 1.5 2 2.5 3

kL

N

o

rm

ali

sed po

w

er

(a)

0 5 10

r�

(dB r

e 1)

−15 −10 −5

0 5 10 15 20 25

kL

(b)

Figure 20: (a) he normalised input power to a inite plate subjected to force and moment excitations at four contact points averaged over

various possible forcing locations and frequency bands: mean (— numerical calculation,− −ininite plate) and mean±standard deviation

(⋅ ⋅ ⋅numerical calculation,− ⋅ −ininite plate). (b) he relative standard deviation of the input power; — numerical calculation,− ⋅ −ininite

0 2 4 6 8 10 12 0

2 4 6 8 10

kL

r�

(dB r

e 1)

−10 −8 −6 −4 −2

(a)

0 10 20 30 40 50 60

kL

0 2 4 6 8 10

r�

(dB r

e 1)

−10 −8 −6 −4 −2

(b)

Figure 21: he relative standard deviation of the input power to a inite plate subjected to force and moment excitations at four contact points

averaged over various possible forcing locations and frequency bands; — numerical calculation and− −(33) (� = 0.01,� = 0.005m, and

� = 0.005; (a)� = 0.07m and (b)� = 0.35m).

� = 0.05. he spatial separation of the contact points is again assumed to form a rectangular shape and � is the length of the diagonal. he results agree well with those from the ininite plate above�� = 10. Below this, as inSection 4.4, the agreement deteriorates due to small damping. his is clearly shown in the relative standard deviation plotted in Figure

20(b). However, it can be seen that the numerical result has a

good agreement with that from the prediction using (33).

Figure 21shows the relative standard deviation for � =

0.01for diferent distances between the contact points. Again, good agreement can be seen using (33), particularly for high frequencies. InFigure 21(b)diferences are seen for�� > 25, but they are less than 1 dB. At low��, the prediction difers by 2 dB on average due to the very low-modal overlap. From the results presented, it shows that (33), which is applicable only for the translational force, can also be used to predict the contribution of moment at high frequencies.

### 7. Conclusions

he uncertainty in input power to a structure due to uncer-tainty in the excitation has been investigated. For an ininite plate, the distance between the location where multiple forces are applied is not important if it is less than half a structural wavelength. he variance of the input power due to uncertainty in excitation phase and location tends to decrease as the nondimensional frequency��increases. For multiple point excitation where the relative phases are random, the mean power and the variance depend only on the input and transfer mobility, respectively.

As for the ininite plate, the variance of the input power to a inite plate also typically decreases as the frequency increases. he frequency average of the input power over all possible forcing locations from multiple contact points can be estimated reasonably and accurately by using the ininite plate result. However, for a very low damping (<5%) the

agreement deteriorates and the simple prediction of the mean and variance can be used [1,17].

he uncertainty in the force amplitude at the contact points has also been discussed. Unless the spatial separation of the excitation locations is small, the distribution of the force amplitude through the contact points is important to obtain accurate estimates of the variation of the mean and standard deviation of the input power, particularly at low frequencies. his variation reduces as the damping loss factor is increased.

he relative efect of moment excitation can be expressed in terms of a force and a distance corresponding to a char-acteristic of the source. It can also be scaled as a function of the input power of the force and the structural wavenumber. his efect tends to increase as frequency increases. he contribution to the total input power can be predicted using the simple expression of the relative standard deviation for the force. However in any event, the efects of moment excitation are typically small at low frequency and in any event are generally less than the efects of force excitation. hey are typically, thus, of secondary importance.

Finally, there remains the moot point of what uncertainty is, in practice, acceptable. his is to a large extent dependent on the typical uncertainty of machinery characterisation methods, such as the reception plate method. he attempt here is to quantify to some extent the uncertainty introduced by some details of the excitation, details that would typically not be measured.

### Appendices

### A. Force and Moment Transfer Mobilities for

### an Infinite plate

Figure 14deines the force-moment excitation directions. he

point at distance�away from the excitation point. he�-axis is perpendicular to the surface of the plate. he mobility terms of an ininite plate structure subjected to a harmonic force or moment point loading are given by [15]

̃�V�

where�is the bending stifness,�is the structural
wavenum-ber, �_{�}(2) is the �th-order Hankel function of the second
kind, and�� is the�th-order modiied Bessel function of
the second kind. he asymptotic forms of the functions for

�� ≫ 1are given by [16]

### B. Force and Moment Mobilities for

### a Finite Plate

For a inite rectangular plate, the mobilities can be written in terms of a modal summation. he point reference (0,0) is located at the corner of the plate. he moment-rotational velocity transfer mobilities at frequency �for a plate with damping loss factor�are given by [15]

and the force-rotational and the moment-translational veloc-ity transfer mobilities are

̃�̇���= ��∑∞

whereΦ��is the mass normalised mode shape and���is the natural frequency of the(�, �)mode as deined in (28) for a simply supported boundary condition. he point mobility can be obtained by setting� = �0and� = �0.

### Acknowledgments

### References

[1] R. S. Langley and A. W. M. Brown, “he ensemble statistics of the energy of a random system subjected to harmonic

excita-tion,”Journal of Sound and Vibration, vol. 275, no. 3–5, pp. 823–

846, 2004.

[2] R. S. Langley and A. W. M. Brown, “he ensemble statistics of

the band-averaged energy of a random system,”Journal of Sound

and Vibration, vol. 275, no. 3–5, pp. 847–857, 2004.

[3] R. S. Langley and V. Cotoni, “Response variance prediction in

the statistical energy analysis of built-up systems,”Journal of the

Acoustical Society of America, vol. 115, no. 2, pp. 706–718, 2004. [4] J. M. Mondot and B. Petersson, “Characterization of structure-borne sound sources: the source descriptor and the coupling

function,”Journal of Sound and Vibration, vol. 114, no. 3, pp.

507–518, 1987.

[5] B. A. T. Petersson and B. M. Gibbs, “Use of the source descriptor concept in studies of multi-point and multi-directional

vibra-tional sources,”Journal of Sound and Vibration, vol. 168, no. 1,

pp. 157–176, 1993.

[6] B. Petersson and J. Plunt, “On efective mobilities in the predic-tion of structure-borne sound transmission between a source structure and a receiving structure, part I: theoretical

back-ground and basic experimental studies,”Journal of Sound and

Vibration, vol. 82, no. 4, pp. 517–529, 1982.

[7] B. Petersson and J. Plunt, “On efective mobilities in the pre-diction of structure-borne sound transmission between a source structure and a receiving structure, part II: procedures

for the estimation of mobilities,”Journal of Sound and Vibration,

vol. 82, no. 4, pp. 531–540, 1982.

[8] R. A. Fulford and B. M. Gibbs, “Structure-borne sound power and source characterisation in multi-point-connected systems,

Part 1: case studies for assumed force distributions,”Journal of

Sound and Vibration, vol. 204, no. 4, pp. 659–677, 1997. [9] R. A. Fulford and B. M. Gibbs, “Structure-borne sound power

and source characterization in multi-point-connected systems.

Part 3: force ratio estimates,”Journal of Sound and Vibration, vol.

225, no. 2, pp. 239–282, 1999.

[10] L. Cremer, M. Heckl, and B. A. T. Petersson,Structure Borne

Sound, Springer, Berlin, Germany, 3rd edition, 2005.

[11] B. M. Gibbs, R. Cookson, and N. Qi, “Vibration activity and mobility of structure-borne sound sources by a reception plate

method,”Journal of the Acoustical Society of America, vol. 123,

no. 6, pp. 4199–4209, 2008.

[12] B. A. T. Petersson and B. M. Gibbs, “Towards a structure-borne

sound source characterization,”Applied Acoustics, vol. 61, no. 3,

pp. 325–343, 2000.

[13] T. A. Evans and A. T. Moorhouse, “Mean and variance of injected structure borne sound power due to missing source activity phase data,” Proceeding of NOVEM, Oxford, UK, 2009.

[14] L. Hinke,Modelling approaches for the low-frequency analysis of

the built-up structures with non- deterministic properties [Ph.D. thesis], University of Southampton, 2008.

[15] M. J. Brennan and P. Gardonio, “Mobility and impedance

meth-ods in structural dynamics,” inChapter 9 in Advanced

Appli-cations in Acoustics, Noise and Vibration, F. J. Fahy and J. G. Walker, Eds., Spon Press, 2004.

[16] M. Danos and J. Rafelski,Pocketbook of Mathematical Functions,

Harri Deutsch; Abridged, 1984.

[17] R. H. Lyon and R. G. DeJong,Application of Statistical Energy

Analysis, Butterworth-Heinemann, 1995.

[18] A. T. Moorhouse, “A dimensionless mobility formulation for

evaluation of force and moment excitation of structures,”

Jour-nal of the Acoustical Society of America, vol. 112, no. 3, pp. 972– 980, 2002.

[19] S. H. Yap and B. M. Gibbs, “Structure-borne sound transmission from machines in buildings, part 2: indirect measurement of force and moment at the machine-receiver interface of a single

point connected system by a reciprocal method,”Journal of

Sound and Vibration, vol. 222, no. 1, pp. 99–113, 1999.

[20] B. A. T. Petersson, “Structural acoustic power transmission by point moment and force excitation, Part II: plate-like

struc-tures,”Journal of Sound and Vibration, vol. 160, no. 1, pp. 67–91,

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